Confinement-Engineered Superconductor to Correlated-Insulator Transition in a van der Waals Monolayer

Transition metal dichalcogenides (TMDC) are a rich family of two-dimensional materials displaying a multitude of different quantum ground states. In particular, d3 TMDCs are paradigmatic materials hosting a variety of symmetry broken states, including charge density waves, superconductivity, and magnetism. Among this family, NbSe2 is one of the best-studied superconducting materials down to the monolayer limit. Despite its superconducting nature, a variety of results point toward strong electronic repulsions in NbSe2. Here, we control the strength of the interactions experimentally via quantum confinement and use low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS) to demonstrate that NbSe2 is in close proximity to a correlated insulating state. This reveals the coexistence of competing interactions in NbSe2, creating a transition from a superconducting to an insulating quantum correlated state by confinement-controlled interactions. Our results demonstrate the dramatic role of interactions in NbSe2, establishing NbSe2 as a correlated superconductor with competing interactions.

: Background removal and fit of SC spectra and estimation of Coulomb gap: (a) Normalized 350 mK and 4 K spectra, taken in 18400 nm 2 island. (b) 350 mK spectrum after removing the 4 K background along with its Dynes fit. (c) Coulomb gap spectra taken in 329 nm 2 island (spectrum at T = 4 K shown for reference. (d) Numerical derivative d 2 I/dV 2 of the 350 mK spectrum (arrows indicate Coulomb gap energies). by dividing the superconducting spectra with the 4K spectra and the resultant spectra was fitted to the Dynes function N S (E) = ( |E|+iΓ(T ) √ (|E|+iΓ(T )) 2 −∆(T ) 2 ) as shown in Fig. S2(b). Here, N S (E) denotes the normalized background subtracted SC density of states, denotes the real part, Γ(T ) is the temperature dependent quasiparticle lifetime broadening parameter, also known as the Dynes parameter and ∆(T ) is the temperature dependent SC energy gap.
The size dependence of the extracted Dynes parameter is shown in Fig. S3.
To extract the Coulomb gap magnitude the raw spectra ( Fig. S2(c)) was numerically differentiated to obtain d 2 I/dV 2 , which has a characteristic peak-dip features in positive and negative biases respectively, as shown by arrows in Fig. S2(d). The average bias voltage location for peak and dip was taken to be the Coulomb gap.
S-3 Figure S3: Size dependence of Dynes parameter from the fit with respect to the size of the SC islands.

Size dependence of CDW and band structure
In Fig. S4(a,c,e), we show the presence of 3 × 3 charge density wave in islands of sizes 480 nm 2 , 1130 nm 2 and 9500 nm 2 respectively as indicated by their respective fast Fourier transforms (FFT) in Fig. S4 (b,d,f). This indicates that the 3 × 3 CDW modulation present in the extended monolayer NbSe 2 survives down to the length scales where correlations are observed. Also, the intensity of the CDW modulation becomes inhomogeneous in the correlated regime. This is a further signature of the strengthening of interactions; perhaps such a change could be associated to an additional charge ordering promoted by interactions. In Fig. S4(g), the large bias dI/dV remains unchanged with island sizes showing the characteristic Nb d-band feature observed in the extended monolayer. All these results clearly demonstrates that the 2H-polytype of NbSe 2 survives down to the length scales where correlations are being observed. S-4

Magnetic field dependence
In Figure S5, we characterise and distinguish the different types of observed spectra by their respective magnetic field dependencies. The representative spectra chosen are superconducting (SC) spectra from island of size 4200 nm 2 ( Fig. S5(a)), Coulomb gapped spectra from island of size 650 nm 2 ( Fig. S5(b)), and proximitized SC spectra from island of size 650 nm 2 (Fig. S5(c)). The Coulomb nature of the island having size 2700 nm 2 becomes evident from the dI/dV taken at magnetic field of 3 T, which is greater than H c2 for the superconducting islands. As shown in Fig. S5(d), the superconducting island having size close to the SC-Coulomb phase boundary (4200 nm 2 ) and larger SC island (8400 nm 2 ) have almost indistinguishable dI/dV at H = 3 T whereas in the 2700 nm 2 island, significantly prominent gap signature is present easily distinguishable from the larger SC islands. The fitted BCS gap magnitude for island of size 4200 nm 2 decreases monotonically to zero with increasing field, whereas Coulomb gap in 650 nm 2 island remains finite and almost constant S-5 ( Fig. S5(e)). The normalized zero bias conductance (ZBC) of both SC and proximitized SC spectra monotonically increases to one with increasing field whereas ZBC for Coulomb gap remains almost constant with increasing field (Fig. S5(f)). Figure S5: Magnetic field dependence of SC gap, Coulomb gap and proximitised SC gap. Magnetic field dependence of dI/dV spectra in island of size (a) 4200 nm 2 including BCS fits up to 1.5 T, (b) 650 nm 2 , (c) 650 nm 2 having proximity to SC island. (d) Comparison of dI/dV taken at 3 T on islands with sizes of 2700, 4200 and 8400 nm 2 . (e), (f) Magnetic field dependence of (e) gap magnitude and (f) normalized zero bias conductance. Spectra in panels (a),(b),(c) are offset vertically for clarity.
It is worthwhile to note that even for large SC islands we still have a small residual gap at high magnetic fields which could be attributed to a correlated pseudogap state that appears when the superconductivity is quenched, and can have additional symmetry broken states of charge or spin-ordering. Precise nature of this pseudogap state remains an open question in the field of correlated superconductors. It is noted that the small residual gap in high fields observed in Pb islands was attributed to the Coulomb gap. S2

S-6
Spatial dependence of point spectra within a single island In Fig. S6(a,b), we demonstrate the representative spectra in edge and middle of a Coulombgapped island indicating that the magnitude of the Coulomb gap remains constant. Spatial variation of SC spectra is demonstrated in Fig. S6(d,g) for 2 different sized islands in Fig. S6(c,f) having areas 10400 nm 2 and 18400 nm 2 , respectively (lateral sizes 102 nm and 136 nm, respectively). It is observed that the edges of the SC islands shows larger SC gap compared to the middle of the islands and this variation is larger for bigger islands.   Superconducting gap map and zero bias conductance map of smallest superconducting island For the smallest island on which the superconducting spectra was observed (size ∼ 4200 nm 2 ), dI/dV map was obtained over an area 12.5 nm × 12.5 nm ( Fig. S7(a)). The gap obtained from the fitted spectra with Dynes model and the normalised zero bias conductance shows spatial variation as seen in Fig. S7(b,c). It is also apparent from the spatial variations that the regions where ZBC is higher, SC gap is lower and vice versa. ZBC vs SC gap plot fit gives a slope of ∼ -0.54 indicating strong anticorrelation. Dependence of the transition on the number of manybody orbitals Here we show that the transition between the correlated gap and the superconducting gap takes place independently on the number of orbitals considered in the calculation. In particular, we show in Fig. S8 the spectral function as a function of the size of the island, taking a different number of many-body orbitals in the calculations. It can be clearly seen that both for 2n = 10 ( Fig. S8(a)) and 2n = 12 ( Fig. S8(b)) orbitals, a transition between a correlated gap to a superconducting one emerges, analogous to the calculations of Fig. 3(a) in the main manuscript. In the absence of U or V there would not be a phase transition between the correlated and superconducting regimes. Nevertheless, in the absence of repulsive interactions, there could still be a phase transition as function of the system size between a single-particle gap coming from quantized energy levels and a superconducting gap. This transition would have an associated smooth evolution of the gap, in contrast with the sharp transition we observe in our data. We finally note that the zero-bias anomaly can be modelled with the model of the dynamical Coulomb blockade. S3 While modelling the dynamical Coulomb blockade in this system could be very interesting, it is significantly beyond the scope of this manuscript. Figure S8: Electronic spectral function as a function of the size of the island, for a different number of many-body orbitals, 2n = 10 in (a) and 2n = 12 in (b). It is observed that the transition between different gaps happens irrespective of the number of orbitals.

SC-Coulomb phase boundary as a function of the strength of the proximity effect
Here we address the dependence of the transition between the correlated and superconducting state driven by proximity. First, it is worth noting that in the absence of superconducting proximity, the NbSe 2 would not show the presence of superconductivity due to its finite nature. However, for intermediate islands, the existence of a small proximity drives the system to the superconducting state. The critical length at which such transition takes place depends on the strength of the proximity effect as shown in Fig. S9. The transition remains sharp for finite proximity effects, illustrating that a sharp transition with system size is expected irrespective of the exact value of the superconducting proximity effect. In the following we will consider an atomistic tight binding model for the Wannier states of the NbSe 2 band, one per Nb atom, sitting in a triangular lattice. S4 The total Hamiltonian takes the form where H kin is the spin-independent hopping term H CDW is the charge density-wave order and H SOC is the intrinsic Ising spin-orbit coupling In the previous terms, t ij is the hopping term that incorporates up to 4th-neighbor hopping, CDW,i are the modulated onsite energies associated to the CDW order, ν ij = ±1 alternate signs between the different bonds leading to a C 3 symmetric hopping, S5 and σ z is With the previous Hamiltonian, we now consider the effect of interactions on a finite island. We include interactions in the form of long-range Coulomb interaction in the atomistic model as where r i is the location of Nb atom i, and V 0 is the Coulomb prefactor in atomic units. As this term turns the system into a full-fledge many-body problem, we will consider the impact of interactions only in the lowest energy states. We project this Coulomb interaction to the states closest to the Fermi surface Ψ α , giving rise to an interaction of the form S-12 where V ijkl are obtained by projecting Eq. 5 into the low energy states Ψ α . We now take finite-size islands with different shapes and different number of atoms, and compute the effective interaction V ijkl . We perform this procedure on islands whose atomistic Hamiltonian has zero and non-zero spin-orbit coupling and zero and non-zero charge density wave, the results are shown in Fig. S11. As the effective interaction V ijkl is a four dimensional tensor, generically complex-valued, we will characterize the strength of the repulsive interactions by the average of its absolute value. As shown in Fig. S11, we observe a robust 1/L behavior of the projected interaction, independently on the presence or absence of Ising spin-orbit coupling and charge density wave. These results demonstrate that not Ising spin-orbit cou- Finally, we comment on interaction screening effects in NbSe 2 . From the theoretical point of view, providing an accurate estimate of the screening would require performing RPA (random phase approximation) density functional theory (DFT) calculations of the dielectric screening, which would account for both intraband and interband screening in the system. We note that such estimate cannot be reliably performed with the low energy tight binding model we are considering, as interband contributions would be completely neglected in that scenario.

Estimation of coherence length
To estimate the superconducting coherence length, we first fitted the proximitized spectra of Fig. 4 (c) with Dynes' model. The extracted gap values as function of distance was then plotted with an exponential decay to extract the Coherence length in Fig. S12(a). The gap, SC gap inside the NbSe 2 island, residual SC gap in HOPG (due to proximity of nearby islands), spatial location of the boundary between NbSe 2 and HOPG and the Coherence length respectively. This fit gives us a Coherence length ξ ≈ 7.3 nm.
Alternatively, the coherence can be determined from the field dependence of the SC island shown in Fig. S12(b). The SC island size here is 8400 nm 2 . From the linear interpolation of the normalized zero bias conductance (ZBC), we can estimate the upper critical field (H C2 ), which comes out to be ≈ 2.47 T. So, the estimated coherence length will be ξ = 1 2π φ 0 H c2 = 11.5 nm.

Spatial dependence of islands in proximity
The proximity induced SC gap in the island in Fig. S13(a) varies spatially as illustrated in Fig. S13(b). We observe the presence of SC order by the dip in the conductance at zero bias and the presence of coherence peaks in the different locations of the proximitised island.
It indicates that the SC order has been established in the entire island. The SC spectra is asymmetric in conductance values at the coherence peak locations. There is however a variation in the asymmetry observed in the individual spectra at different locations. The histogram of the asymmetry defined by the (normalised conductance at -ve coherence peak location)-(normalised conductance at +ve coherence peak location) shows a distribution asymmetric about zero (mean value of 0.11 from Gaussian fit). The proximitised non-SC island in Fig. S13(d) have variations in the local spectra as seen from Fig. S13(e). Here, the strong electron-hole asymmetries are typical of a non-superconducting origin of the gap, S6 and support the strongly correlated nature of the small islands.